That is indeed a HDD, and it appears that this specific model is known for having some relatively strange behaviors.
How important are these files to you? I ask because there are a few possibilities in regards to what is going on, and it is possible that the only way of retrieving these files is through professional data recovery - further, if this drive has suffered from certain types of failures, continuing to troubleshoot carries the risk of making the problem worse. If the files are so important that they must absolutely be retrieved no matter what the cost, then not trying anything else with the hard drive, setting it aside, and enlisting the services of a respectable data recovery firm is probably the best course of action. Price-wise you could be looking at anything from $250-2000+. Most firms will evaluate a drive for free and provide you with more concrete pricing before beginning the recovery process. For what it is worth, with the data recovery firm we use at my work, most hardware recovery operations tend to be between $400-600.
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If you do NOT want to pursue professional data recovery services, or you want to try more troubleshooting before exploring that option (with the understanding that this could complicate the recovery process and potentially increase the price) the next step would be to determine whether this drive has suffered from hardware failure or corruption (or potentially both.) If this is an ExFAT drive, considering how the drive did NOT suffer a fall but did suffer an unexpected power loss, my inclination is that this is a corruption issue (if I misunderstood you and the drive was exposed to a fall or impact, that is a different story.)
To try to determine if the drive is still physically functional, I would personally try a few things if this were me. First, when I plug the drive in, I would put my hand on top of the enclosure to determine whether or not I can feel the vibration that is caused from the drive 'spinning' up (this sound is also audible if you put your ear next to it.) Further, I would listen to the drive to make sure there was no clicking or clacking sounds, and to also verify that the drive is not spinning up, spinning down, and continuing to spin up and down every few seconds. Provided the drive spins up and is not making unusual noises, I would then use a SMART program to try to pull data on the drive health. The App I use for this is called
DriveDX and there is a free trial - it's easy to install, use, and understand. Provided that DriveDX can 'see' the drive and that the health status comes back as 'good', at this point I would assume that the drive is probably not suffering from hardware failure and that this is corruption.
The next step we could try is using the Terminal and About This Mac to see if either can 'see' the drive (even though the Disk Utility cannot,) and try to repair the drive from the Terminal with the fsck function if the Mac's system profiler can still see the drive/path and we can figure out the file system. If this is an ExFAT drive, I have found that the ChkDsk /f function on a Windows computer is extremely effective at repairing corruption damage to ExFAT volumes after a power loss incident on a Mac where afterwards the Mac can no longer 'see' the drive. (When this happens to my ExFAT volumes, as soon as I plug the drive into a Windows 10 computer I get a message telling me that the drive is damaged and am asked whether or not I want to let Windows repair the drive - after telling it OK, about 15-45 minutes later it finishes the repair and has been successful in most cases.)
Don't act on this yet until others have chimed in! I'm far from an expert on this topic, so other Members may have some thoughts on this proposed course of action, perhaps someone like
@Weaselboy or
@Queen6 ?
